Henry Ernest Dudeney/Modern Puzzles/Geometrical Problems/Various Geometrical Puzzles
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Henry Ernest Dudeney: Modern Puzzles: Geometrical Problems: Various Geometrical Puzzles
$124$ - Drawing a Straight Line
- If we want to describe a circle we use an instrument that we call a pair of compasses,
- but if we need a straight line we use no such instrument --
- we employ a ruler or other straight edge.
- In other words, we first seek a straight line to produce our required straight line,
- which is equivalent to using a coin, saucer of other circular object to draw a circle.
- Now, imagine yourself in such a position that you cannot obtain a straight edge --
- not even a piece of thread.
- Could you devise a simple instrument that would draw your straight line,
$125$ - Making a Pentagon
- How do you construct a regular pentagon on a given unit straight line?
$126$ - Drawing an Oval
- It is well-known that you can draw an ellipse by sticking two pins into the paper, enclosing them with a loop of thread,
- and keeping the loop taut, running a pencil all the way round till you get back to the starting point.
- Suppose you want an ellipse with a given major axis and minor axis.
- How do you arrange the position of the pins, and what would be the length of the thread?
$127$ - With Compasses Only
$128$ - Lines and Squares
- With how few straight lines can you make exactly one hundred squares?
- Thus, in the first diagram it will be found that with nine straight lines I have made twenty squares
$129$ - The Circle and Discs
- During a recent visit to a fair we saw a man with a table,
- The circular discs were all of the same size, and each, of course, smaller than the red circle.
- he showed that it was "quite easy when you know how," by covering up the circle himself without any apparent difficulty,
- but many tried over and over again and failed every time.
- It was a condition that when once you had placed any disc you were not allowed to shift it,
- otherwise, by sliding them about after they had been placed, it might be tolerably easy to do.
- Let us assume that the red circle is six units in diameter.
- Now, what is the smallest possible diameter for the five discs in order to make a solution possible?
$130$ - Mr. Grindle's Garden
- "My neighbour," said Mr. Grindle, "generously offered me, for a garden,
- "And what was the largest area you were able to enclose?" asked his friend.
- Perhaps the reader can discover Mr. Grindle's correct answer.
$131$ - The Garden Path
- A man has a rectangular garden, $55$ yards by $40$ yards,
- and he makes a diagonal path, one yard wide, exactly in the manner indicated in the diagram.
- What is the area of the path?
$132$ - The Garden Bed
- A man has a triangular lawn of the proportions shown,
- and he wants to make the largest possible rectangular flower-bed without enclosing the tree.
$133$ - A Problem for Surveyors
- A man bought a little field, and here is a scale map that was given to me.
- I asked my surveyor to tell me the area of the field,
- but he said it was impossible without some further measurements;
- the mere length of one side, $7$ rods, was insufficient.
- What was his surprise when I showed him in about two minutes what was the area!
- Can you tell how it is to be done?
$134$ - A Fence Problem
- A man has a square field, $60 \ \mathrm {ft.}$ by $60 \ \mathrm {ft.}$, with other property, adjoining the highway.
- For some reason he put up a straight fence in the line of three trees, as shown,
- and the length of fence from the middle tree to the tree on the road was just $91$ feet.
- What is the distance in exact feet from the middle tree to the gate on the road?
$135$ - The Domino Swastika
- Form a square frame with twelve dominoes, as shown in the illustration.
- Now, with only four extra dominoes, form within the frame a swastika.
$136$ - A New Match Puzzle
- I have a box of matches.
- I find that I can form with them any given pair of these four regular figures, using all the matches every time.
- This, if there were eleven matches, I could form with them, as shown, the triangle and pentagon
- Of course there must be the same number of matches in every side of a figure.
- Now, what is the smallest number of matches I can have in the box?
$137$ - Hurdles and Sheep
- A farmer says that four of his hurdles will form a square enclosure just sufficient for one sheep.
- That being so, what is the smallest number of hurdles that he will require for enclosing ten sheep?
$138$ - The Four Draughtsmen
- The four draughtsmen are shown exactly as they stood on a square chequered board --
- not necessarily eight squares by eight --
- but the ink with which the board was drawn was evanescent,
- so that all the diagram except the men has disappeared.
- How many squares were there in the board and how am I to reconstruct it?
- I know that each man stood in the middle of a square,
- one on the edge of each side of the board and no man in a corner.
$139$ - A Crease Problem
- Fold a page, so that the bottom outside corner touches the inside edge and the crease is the shortest possible.
$140$ - The Four-Colour Map Theorem
- In colouring any map under the condition that no contiguous countries shall be coloured alike,
- not more than four colours can ever be necessary.
- Countries only touching at a point ... are not contiguous.
- I will give, in condensed form, a suggested proof of my own
- which several good mathematicians to whom I have shown it accept it as quite valid.
- Two others, for whose opinion I have great respect, think it fails for a reason that the former maintain will not "hold water".
- The proof is in a form that anybody can understand.
- It should be remembered that it is one thing to be convinced, as everybody is, that the thing is true,
- but quite another to give a rigid proof of it.
$141$ - The Six Submarines
- If five submarines, sunk on the same day, all went down at the same spot where another had previously been sunk,
- how might they all lie at rest so that every one of the six U-boats should touch every other one?
- To simplify we will say, place six ordinary wooden matches so that every match shall touch every other match.
- No bending or breaking allowed.
$142$ - Economy in String
- Owing to the scarcity of string a lady found herself in this dilemma.
- In making up a parcel for her son, she was limited to using $12$ feet of string, exclusive of knots,
- which passed round the parcel once lengthways and twice round its girth, as shown in the illustration.
- What was the largest rectangular parcel that she could make up, subject to these conditions?
$143$ - The Stone Pedestal
- In laying the base and cubic pedestal for a certain public memorial,
- There was exactly the same number of these blocks (all uncut) in the pedestal as in the square base on the centre of which it stood.
- Look at the sketch and try to determine the total number of blocks actually used.
- The base is only a single block in depth.
$144$ - The Bricklayer's Task
- When a man walled in his estate, one of the walls was partly level and partly over a small rise or hill,
- precisely as shown in the drawing herewith, wherein it will be observed that the distance from $A$ to $B$ is the same as from $B$ to $C$.
- Now, the master-builder desired and claimed that he should be paid more for the part that was on the hill than for the part that was level,
- since (at least, so he held) it demanded the use of more material.
- But the employer insisted that he should pay less for that part.
- It was a nice point, over which they nearly had recourse to the law.
- Which of them was in the right?
$145$ - A Cube Paradox
- I had two solid cubes of lead, one very slightly larger than the other.
- Through one of them I cut a hole (without destroying the continuity of the four sides)
- so that the other cube could be passed right through it.
- On weighing them afterwards it was found that the larger cube was still the heavier of the two.
- How was this possible?
$146$ - The Cardboard Box
- If I have a closed cubical cardboard box, by running the penknife along seven of the twelve edges (it must always be seven)
- I can lay it out in one flat piece in various shapes.
- Thus, in the diagram, if I pass the knife along the darkened edges and down the invisible edge indicated by the dotted line, I get the shape $A$.
- Another way of cutting produces $B$ or $C$.
- It will be seen that $D$ is simply $C$ turned over, so we will not call that a different shape.
- Now, how many shapes can be produced?
$147$ - The Austrian Pretzel
- Here is a twisted Vienna bread roll, known as a Pretzel.
- The twist, like the curl in a pig's tail, is entirely for ornament.
- The Wiener Pretzel, like some other things, is doomed to be cut up or broken, and the interest lies in the number of resultant pieces.
- Suppose you had the Pretzel depicted in the illustration lying on the table before you,
- what is the greatest number of pieces into which you could cut it with a single straight cut of a knife?
- In what direction would you make the cut?
$148$ - Cutting the Cheese
- I have a piece of cheese in the shape of a cube.
- How am I to cut it in two pieces with one straight cut of the knife
- so that the two new surfaces produced by the cut shall each be a perfect hexagon?
$149$ - A Tree-Planting Puzzle
- How do you plant $13$ trees so as to form $9$ straight rows of $4$ trees each?